Provides an interface for a smoothed version of the worst-case scenario risk measure using the expectation risk quadrangle. More...

#include <ROL_SmoothedWorstCaseQuadrangle.hpp>

Inheritance diagram for ROL::SmoothedWorstCaseQuadrangle< Real >:

Public Member Functions
	SmoothedWorstCaseQuadrangle (const Real eps)
	Constructor. More...

	SmoothedWorstCaseQuadrangle (ROL::ParameterList &parlist)
	Constructor. More...

Real	error (Real x, int deriv=0)
	Evaluate the scalar error function at x. More...

Real	regret (Real x, int deriv=0)
	Evaluate the scalar regret function at x. More...

void	check (void)
	Run default derivative tests for the scalar regret function. More...

Public Member Functions inherited from ROL::ExpectationQuad< Real >
virtual	~ExpectationQuad (void)

	ExpectationQuad (void)

Private Member Functions
void	parseParameterList (ROL::ParameterList &parlist)

void	checkInputs (void) const

Private Attributes
Real	eps_

Detailed Description

template<class Real>
class ROL::SmoothedWorstCaseQuadrangle< Real >

Provides an interface for a smoothed version of the worst-case scenario risk measure using the expectation risk quadrangle.

The worst-case scenario risk measure is

\[ \mathcal{R}(X) = \sup_{\omega\in\Omega} X(\omega). \]

$\mathcal{R}$ is a law-invariant coherent risk measure. Clearly, $\mathcal{R}$ is not differentiable. As such, this class defines a smoothed version of $\mathcal{R}$ the expectation risk quadrangle. In the nonsmooth case, the scalar regret function is $v(x) = 0$ if $x \le 0$ and $v(x) = \infty$ if $x > 0$. Similarly, the scalar error function is $e(x) = -x$ if $x \le 0 $ and $e(x) = \infty$ if $x > 0$. To smooth $\mathcal{R}$, we perform Moreau-Yosida regularization on the scalar error function, i.e.,

\[ e_\epsilon(x) = \inf_{y\in\mathbb{R}} \left\{ e(y) + \frac{1}{2\epsilon} (x-y)^2\right\} % = \left\{\begin{array}{l l} % -\left(x+\frac{\epsilon}{2}\right) & % \text{if \f$x \le -\epsilon\f$}\\ % \frac{1}{2\epsilon}x^2 & \text{if \f$x > -\epsilon\f$}. % \end{array}\right. \]

for $\epsilon > 0$. The corresponding scalar regret function is $v_\epsilon(x) = e_\epsilon(x) + x$. $\mathcal{R}$ is then implemented as

\[ \mathcal{R}(X) = \inf_{t\in\mathbb{R}}\left\{ t + \mathbb{E}[v_\epsilon(X-t)] \right\}. \]

ROL implements this by augmenting the optimization vector $x_0$ with the parameter $t$, then minimizes jointly for $(x_0,t)$.

Definition at line 89 of file ROL_SmoothedWorstCaseQuadrangle.hpp.

Constructor & Destructor Documentation

template<class Real >

ROL::SmoothedWorstCaseQuadrangle< Real >::SmoothedWorstCaseQuadrangle ( const Real eps )

inline

Constructor.

Parameters

[in] eps is the regularization parameter

Definition at line 123 of file ROL_SmoothedWorstCaseQuadrangle.hpp.

References ROL::SmoothedWorstCaseQuadrangle< Real >::checkInputs().

template<class Real >

ROL::SmoothedWorstCaseQuadrangle< Real >::SmoothedWorstCaseQuadrangle ( ROL::ParameterList & parlist )

inline

Constructor.

Parameters

[in] parlist is a parameter list specifying inputs

parlist should contain sublists "SOL"->"Risk Measure"->"Smoothed Worst-Case Quadrangle" and within the "Smoothed Worst-Case Quadrangle" sublist should have the following parameters